474 research outputs found
Bayesian inference for inverse problems
Traditionally, the MaxEnt workshops start by a tutorial day. This paper
summarizes my talk during 2001'th workshop at John Hopkins University. The main
idea in this talk is to show how the Bayesian inference can naturally give us
all the necessary tools we need to solve real inverse problems: starting by
simple inversion where we assume to know exactly the forward model and all the
input model parameters up to more realistic advanced problems of myopic or
blind inversion where we may be uncertain about the forward model and we may
have noisy data. Starting by an introduction to inverse problems through a few
examples and explaining their ill posedness nature, I briefly presented the
main classical deterministic methods such as data matching and classical
regularization methods to show their limitations. I then presented the main
classical probabilistic methods based on likelihood, information theory and
maximum entropy and the Bayesian inference framework for such problems. I show
that the Bayesian framework, not only generalizes all these methods, but also
gives us natural tools, for example, for inferring the uncertainty of the
computed solutions, for the estimation of the hyperparameters or for handling
myopic or blind inversion problems. Finally, through a deconvolution problem
example, I presented a few state of the art methods based on Bayesian inference
particularly designed for some of the mass spectrometry data processing
problems.Comment: Presented at MaxEnt01. To appear in Bayesian Inference and Maximum
Entropy Methods, B. Fry (Ed.), AIP Proceedings. 20pages, 13 Postscript
figure
Penalized maximum likelihood for multivariate Gaussian mixture
In this paper, we first consider the parameter estimation of a multivariate
random process distribution using multivariate Gaussian mixture law. The labels
of the mixture are allowed to have a general probability law which gives the
possibility to modelize a temporal structure of the process under study. We
generalize the case of univariate Gaussian mixture in [Ridolfi99] to show that
the likelihood is unbounded and goes to infinity when one of the covariance
matrices approaches the boundary of singularity of the non negative definite
matrices set. We characterize the parameter set of these singularities. As a
solution to this degeneracy problem, we show that the penalization of the
likelihood by an Inverse Wishart prior on covariance matrices results to a
penalized or maximum a posteriori criterion which is bounded. Then, the
existence of positive definite matrices optimizing this criterion can be
guaranteed. We also show that with a modified EM procedure or with a Bayesian
sampling scheme, we can constrain covariance matrices to belong to a particular
subclass of covariance matrices. Finally, we study degeneracies in the source
separation problem where the characterization of parameter singularity set is
more complex. We show, however, that Inverse Wishart prior on covariance
matrices eliminates the degeneracies in this case too.Comment: Presented at MaxEnt01. To appear in Bayesian Inference and Maximum
Entropy Methods, B. Fry (Ed.), AIP Proceedings. 11pages, 3 Postscript figure
Information and Covariance Matrices for Multivariate Burr III and Logistic distributions
Main result of this paper is to derive the exact analytical expressions of
information and covariance matrices for multivariate Burr III and logistic
distributions. These distributions arise as tractable parametric models in
price and income distributions, reliability, economics, populations growth and
survival data. We showed that all the calculations can be obtained from one
main moment multi dimensional integral whose expression is obtained through
some particular change of variables. Indeed, we consider that this calculus
technique for improper integral has its own importance in applied probability
calculus.Comment: submitted to Communications in Statistic
An alternative inference tool to total probability formula and its applications
Total probability and Bayes formula are two basic tools for using prior
information in the Bayesian statistics. In this paper we introduce an
alternative tool for using prior information. This new toold enables us to
improve some traditional results in statistical inference. However, as far as
the authors know, there is no work on this subject, except [1]. The results of
this paper can be extended to other branches of probability and statistics. In
Section 2 total probability formula based on median is defined and its basic
properties are proved. A few applications of this new tool are given in Section
3.Comment: Presented at the 23th Int. worskhop on Bayesian and Maximum Entropy
methods (MaxEnt23), Aug. 3-7, 2003, Jackson Hole, US
- …